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 |  موضوع: كتاب Frequency Analysis of Vibration Energy Harvesting Systems الإثنين 25 أكتوبر 2021, 2:05 am | |
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أحضرت لكم كتاب
Frequency Analysis of Vibration Energy Harvesting Systems
Xu Wang
School of Engineering
College of Science Engineering and Health
RMIT University
و المحتوى كما يلي :
List of Figures
Figure 1.1 A single degree of freedom spring-mass-dashpot system
driven by (A) direct force and (B) inertial force. 3
Figure 1.2 Prediction of the mass relative displacement response
from the base displacement excitation using Matlab Simulink
transfer function method. 5
Figure 1.3 Prediction of the relative displacement response from the
base acceleration excitation input using the Matlab Simulink
integration method: (A) Integration schematic of Eq. (1.12);
(B) Matlab Simulink schematic following Eq. (1.12). 6
Figure 1.4 Schematic of the Matlab Simulink state space method
for prediction of the relative displacement response from
the base displacement excitation. 8
Figure 1.5 A sine wave excitation signal input with a frequency
of 275 Hz and RMS acceleration of 9.8 m/s2. 9
Figure 1.6 Parameter inputs of sine wave excitation acceleration
for the source block of the Matlab Simulink. 10
Figure 1.7 The time trace of the relative displacement response. 11
Figure 1.8 A Matlab code for calculation of the relative displacement
frequency response function amplitude versus frequency curve. 11
Figure 1.9 Relative displacement frequency response function
(the relative displacement amplitude over the excitation acceleration
amplitude). 12
Figure 2.1 Schematic of single degree of freedom piezoelectric
vibration energy harvester system connected to a single electric
load resistor (A) the harvester oscillator electromechanical system
(B) energy harvesting circuit. 16
Figure 2.2 Simulation schematic for Eq. (2.10) in Matlab Simulink
with a sine wave base excitation input and a sinusoidal
voltage output at a frequency using the transfer function method. 20
Figure 2.3 (A) Relative acceleration integrated to the relative
displacement; (B) Derivative of the output voltage integrated
to the output voltage; (C) Simulation schematic for Eq. (2.11)
with a sine wave base excitation input and a sinusoidal
voltage output at a frequency using the integration method. 22
Figure 2.4 The Matlab Simulink state space method schematic
for the prediction of the response relative displacement and
output voltage from the base excitation acceleration. 24
Figure 2.5 Output sinusoidal voltage signal (volt) from an input
excitation acceleration signal with a root mean square
acceleration value of 1 g (9.8 m/s2) at an excitation
frequency of 275 Hz. 25
Figure 2.6 The output voltage amplitude versus the base excitation
acceleration amplitude at an excitation frequency of 275 Hz. 26
xiFigure 2.7 Harvested resonant power versus the base excitation
acceleration amplitude at an excitation frequency of 275 Hz. 27
Figure 2.8 The output voltage amplitude versus the mechanical
damping under an input excitation acceleration signal
of an RMS value of 1 g (9.8 m/s2) at an excitation frequency of 275 Hz. 27
Figure 2.9 Harvested resonant power versus the mechanical
damping under an input excitation acceleration signal of an
RMS value of 1 g (9.8 m/s2) at an excitation frequency of 275 Hz. 28
Figure 2.10 The output voltage amplitude versus the electrical load
resistance under an input excitation acceleration signal
of an RMS value of 1 g (9.8 m/s2) at an excitation
frequency of 275 Hz. 28
Figure 2.11 Harvested resonant power versus the electrical load
resistance under an input excitation acceleration signal
of an RMS value of 1 g (9.8 m/s2) at an excitation
frequency of 275 Hz. 29
Figure 2.12 Harvested resonant power versus the force factor
under an input excitation acceleration signal of an RMS
value of 1 g (9.8 m/s2) at an excitation frequency of 275 Hz. 29
Figure 2.13 Matlab codes for simulation of the frequency response
functions of the output relative displacement and voltage
over the input excitation acceleration. 30
Figure 2.14 Amplitude of the relative displacement frequency response
functiondthe output relative displacement divided by the input
excitation acceleration. 31
Figure 2.15 Amplitude of the voltage frequency response
functiondthe output voltage divided by the input excitation
acceleration. 32
Figure 2.16 Harvested resonant power and output voltage versus frequency. 34
Figure 2.17 Dimensionless harvested resonant power versus
dimensionless resistance and force factors for the
single degree of freedom system connected to a load resistor. 36
Figure 2.18 Resonant energy harvesting efficiency versus
dimensionless resistance and force factors for the
single degree of freedom system connected to a load resistor. 38
Figure 3.1 Extraction and storage interface circuits for vibration
energy harvesters, (A) standard; (B) synchronous electric
charge extraction (SECE); (C) parallel synchronous switch
harvesting on inductor (SSHI); (D) series SSHI Circuit. 44
Figure 3.2 Working principle of a full cycle of bridge rectification.
(A) Positive half-cycle; (B) negative half-cycle; (C) positive output
waveform. 45
Figure 3.3 The resonant energy harvesting efficiency versus dimensionless
resistance and force factors for the single degree of freedom
piezoelectric harvester connected to the four types of
xii List of Figuresinterface circuits. (A) Standard; (B) synchronous electric
charge extraction; (C) parallel synchronous switch harvesting
on inductor (SSHI); (D) series SSHI. 56
Figure 3.4 The dimensionless harvested resonant power versus
dimensionless resistance and force factors for the single
degree of freedom piezoelectric harvester connected to
the four types of interface circuits. (A) Standard; (B) synchronous
electric charge extraction; (C) parallel synchronous switch
harvesting on inductor (SSHI); (D) series SSHI. 57
Figure 3.5 Output voltage and harvested resonant power versus the
input excitation acceleration. (A) Harvested resonant power
versus the excitation acceleration; (B) output voltage versus
the excitation acceleration. 61
Figure 3.6 (A) Output voltage and (B) harvested resonant power
versus mechanical damping under the base acceleration
of 9.8 m/s2. 62
Figure 3.7 Output voltage (A) and harvested resonant power (B) versus
electric resistance under the base acceleration of 9.8 m/s2. 63
Figure 3.8 Harvested resonant power versus force factor under the base
acceleration of 9.8 m/s2. 63
Figure 4.1 Schematic of a single degree of freedom electromagnetic
vibration energy harvester connected to a single load resistor. 71
Figure 4.2 Simulation schematic for Eq. (4.9) in Matlab Simulink with a
sine wave base excitation input and a sinusoidal voltage
output at a frequency using the transfer function method. 72
Figure 4.3 (A) Relative acceleration integrated to the relative displacement.
(B) Derivative of the output voltage integrated to the output voltage.
(C) Simulation schematic for Eq. (4.10) with a sine wave base excitation
input and a sinusoidal voltage output at a frequency using the time
domain integration method. 73
Figure 4.4 Matlab codes for simulation of the frequency response functions of
the output relative displacement and voltage over the base input
excitation acceleration. 77
Figure 4.5 Resonant energy harvesting efficiency versus dimensionless
resistance and equivalent force factors for the single degree of
freedom electromagnetic harvester connected to a load resistor. 81
Figure 4.6 Dimensionless harvested resonant power versus dimensionless
resistance and equivalent force factors for the single degree of
freedom electromagnetic harvester connected to a load resistor. 82
Figure 4.7 The resonant energy harvesting efficiency versus dimensionless
resistance and force factors for the single degree of freedom
piezoelectric harvester connected to the four types of interface circuits.
(A) Standard interface; (B) synchronous electric charge extraction;
(C) parallel synchronous switch harvesting on inductor (SSHI);
(D) series SSHI. 96
List of Figures xiiiFigure 4.8 The dimensionless harvested resonant power versus dimensionless
resistance and force factors for the SDOF electromagnetic harvester
connected to the four types of interface circuits. (A) Standard
interface; (B) synchronous electric charge extraction;
(C) parallel synchronous switch harvesting on inductor (SSHI);
(D) series SSHI. 101
Figure 5.1 Dimensionless resonant energy harvesting efficiency of piezoelectric
and electromagnetic vibration energy harvesters versus dimensionless
resistance and force factors for the single degree of freedom
harvesters connected to a load resistance; (A) piezoelectric;
(B) electromagnetic. 109
Figure 5.2 Dimensionless harvested resonant power versus dimensionless
resistance and force factors for the single degree of freedom
piezoelectric and electromagnetic harvesters connected to a load
resistor; (A) piezoelectric; (B) electromagnetic. 111
Figure 5.3 Dimensionless harvested resonant power versus the dimensionless
force factor by fixing the dimensionless resistance of 0.5. 113
Figure 5.4 Dimensionless harvested resonant power versus the dimensionless
force factor by fixing the dimensionless resistance of 1.5. 113
Figure 5.5 Dimensionless harvested resonant power versus the dimensionless
resistance by fixing the dimensionless force factor of 0.5. 114
Figure 5.6 Dimensionless harvested resonant power versus the dimensionless
resistance by fixing the dimensionless force factor of 1.5. 114
Figure 6.1 Schematic of a two degree of freedom piezoelectric vibration energy
harvesting system model. 125
Figure 6.2 The dimensionless harvested resonant power and energy harvesting
efficiency versus various mass ratio (MR ¼ m2/m1). 132
Figure 6.3 The dimensionless harvested resonant power and energy harvesting
efficiency versus various stiffness ratio (KR ¼ k2/k1). 133
Figure 6.4 Case study of a quarter vehicle suspension model with piezoelectric
element insert. 133
Figure 6.5 Simulation schematic for Eqs. (6.1) and (6.2) with a sine wave base
excitation input and a sinusoidal voltage output at a frequency
using the integration method. 135
Figure 6.6 Output voltage for the input excitation acceleration amplitude of
1 g (9.80 m/s2) and excitation frequency of 1.45 Hz. 135
Figure 6.7 Output power for the input excitation acceleration amplitude of 1 g
(9.80 m/s2) and excitation frequency of 1.45 Hz. 136
Figure 6.8 Displacement amplitude ratios of mass 1 and mass 2 with respect to
the input displacement amplitude versus frequency. 137
Figure 6.9 Output voltage and harvested resonant power versus the input
excitation acceleration amplitude. 137
Figure 6.10 The output voltage and harvested resonant power versus frequency. 138
Figure 6.11 Output voltage and harvested resonant power versus electric
load resistance. 139
xiv List of FiguresFigure 6.12 Output voltage and harvested resonant power versus the
wheeletire damping. 140
Figure 6.13 Output voltage and harvested resonant power versus the suspension
damping. 141
Figure 6.14 Output voltage and harvested resonant power versus the force
factor. 142
Figure 6.15 Output voltage versus frequency for various wheeletire mass. 143
Figure 6.16 Output voltage versus frequency for various quarter vehicle masses. 144
Figure 6.17 Output voltage versus frequency for various wheeletire stiffness. 145
Figure 6.18 Output voltage versus frequency for various suspension stiffness. 145
Figure 6.19 The dimensionless harvested resonant power versus the stiffness
ratio (k2/k1). 147
Figure 6.20 Output voltage versus frequency for various wheeletire damping
coefficients. 148
Figure 6.21 Output voltage versus frequency for various suspension damping
coefficients. 148
Figure 6.22 Dimensionless harvested resonant power versus the damping ratio
(c1/c2). 149
Figure 7.1 Schematic of a two degree of freedom piezoelectric vibration energy
harvester inserted with two piezoelectric patch elements. 157
Figure 7.2 The difference of the two dimensionless resonant frequencies versus
the mass ratio M1 and dimensionless frequency ratio U1 under the
synchronous changes of the coupling strength of the piezoelectric
patches. (A) Lz12 1 ¼ Lz22 2 ¼ 0:02; (B) Lz12 1 ¼ Lz22 2 ¼ 5; (C) Lz12 1 ¼ Lz22 2 ¼ 10;
and (D) L2 1
z1 ¼
L2
2
z2 ¼ 40. 162
Figure 7.3 The difference of the two dimensionless resonant frequencies
versus the mass ratio M1 and the U1 under the coupling
strength changes of the primary and auxiliary oscillator systems.
(A) L2 1
z1 ¼ 0:02 Lz22 2 ¼ 5; (B) Lz12 1 ¼ 0:02 Lz22 2 ¼ 10; (C) Lz12 1 ¼ 0:02 Lz22 2 ¼ 40;
and (D) L2 1
z1 ¼ 40 Lz22 2 ¼ 0:02. 163
Figure 7.4 The dimensionless harvested power of the two degree of freedom
piezoelectric vibration energy harvester versus the dimensionless
frequency for different dimensionless mass ratio (M1). (A) The
dimensionless harvested power of the first piezo patch element;
(B) the dimensionless harvested power of the second piezo patch
element; and (C) the total dimensionless harvested power of the first
and second piezo patch elements. DOF, degree of freedom. 164
Figure 7.5 The dimensionless harvested power of the two degree of freedom
piezoelectric vibration energy harvester versus the dimensionless
frequency for different U1. (A) The dimensionless harvested power of
the first piezo patch element; (B) the dimensionless harvested power
of the second piezo patch element; and (C) the total dimensionless
harvested power. DOF, degree of freedom. 165
List of Figures xvFigure 7.6 Dimensionless harvested power of the two degree of freedom
piezoelectric vibration energy harvester versus F and z1.
(A) Dimensionless harvested power of the first piezo patch element;
(B) dimensionless harvested power of the second piezo patch
element; and (C) total dimensionless harvested power. 166
Figure 7.7 Dimensionless harvested power of the two degree of freedom
piezoelectric vibration energy harvester versus F and z2.
(A) dimensionless harvested power of the first piezo patch element;
(B) dimensionless harvested power of the second piezo patch
element; and (C) total dimensionless harvested power. 167
Figure 7.8 The energy harvesting efficiency of the first piezoelectric patch
element versus F and M1 for different coupling strengths.
(A) L2 1
z1 ¼
L2
2
z2 ¼ 0:02; (B) Lz12 1 ¼ Lz22 2 ¼ 5; (C) Lz12 1 ¼ Lz22 2 ¼ 10; and
(D) L2 1
z1 ¼
L2
2
z2 ¼ 40. 168
Figure 7.9 The energy harvesting efficiency of the second piezoelectric patch
element versus F and M1 for different coupling strengths.
(A) L2 1
z1 ¼
L2
2
z2 ¼ 0:02; (B) Lz12 1 ¼ Lz22 2 ¼ 5; (C) Lz12 1 ¼ Lz22 2 ¼ 10; and
(D) L2 1
z1 ¼
L2
2
z2 ¼ 40. 169
Figure 7.10 Schematic of a three degree of freedom piezoelectric vibration
energy harvester inserted with three piezoelectric patch elements. 170
Figure 7.11 The dimensionless harvested power of the 3DOF piezoelectric
vibration energy harvester versus the dimensionless frequency for
different mass ratio M1. (A) Dimensionless harvested power of the
first piezo patch element; (B) dimensionless harvested power of the
second piezo patch element; (C) dimensionless harvested power of
the third piezo patch element; and (D) total dimensionless
harvested power. DOF, degree of freedom. 172
Figure 7.12 The dimensionless harvested power of the 3DOF system versus the
dimensionless frequency for different U1. (A) dimensionless
harvested power of the first piezo patch element; (B) dimensionless
harvested power of the second piezo patch element; (C)
dimensionless harvested power of the third piezo patch element;
and (D) total dimensionless harvested power. DOF, degree of freedom. 173
Figure 7.13 Dimensionless harvested power of the 3DOF piezoelectric vibration
energy harvester versus F and z1. (A) dimensionless harvested power
of the first piezo patch element. (B) dimensionless harvested power of
the second piezo patch element. (C) dimensionless harvested
power of the third piezo patch element. (D) total dimensionless
harvested power. DOF, degree of freedom. 174
Figure 7.14 Dimensionless harvested power of the 3DOF piezoelectric
vibration energy harvester versus F and z2. (A) dimensionless
harvested power of the first piezo patch element. (B) dimensionless
harvested power of the second piezo patch element.
xvi List of Figures(C) dimensionless harvested power of the third piezo patch element.
(D) total dimensionless harvested power. DOF, degree of
freedom. 175
Figure 7.15 The harvested efficiency of the 3DOF piezoelectric vibration energy
harvester versus M1 and F. (A) the efficiency of the first piezo
patch element. (B) the efficiency of the second piezo patch element.
(C) The efficiency of the third piezo patch element. (D) Total
efficiency. DOF, degree of freedom. 175
Figure 7.16 Schematic of a generalized multiple degree of freedom piezoelectric
vibration energy harvester with piezoelectric elements between all
two nearby oscillators. 177
Figure 7.17 The dimensionless harvested power and the harvested power
density versus the number of degrees of freedom of piezoelectric
vibration energy harvester (PVEH). 179
Figure 8.1 A cantilevered bimorph beam clamped by washers with a nut
mass glued at the free end. 188
Figure 8.2 The bimorph cantilevered beam set up on the shaker for lab testing. 189
Figure 8.3 Polytec Laser Doppler Vibrometer system display. 190
Figure 8.4 The measured vibration spectrum and first natural frequency
of 24.375 Hz for the cantilevered beam under a white noise
random force excitation. 191
Figure 8.5 The predicted and measured voltage output versus excitation
frequency for the PZT-5H cantilevered beam. 192
Figure 8.6 The predicted and measured harvested resonant power versus the
excitation frequency for the PZT-5H cantilevered beam. 192
Figure 8.7 The predicted and measured resonant output voltage versus the
external electric load resistance for the PZT-5H cantilevered
beam. 193
Figure 8.8 The predicted and measured resonant output voltages versus the
excitation acceleration amplitude for the PZT-5H cantilevered beam. 194
Figure 8.9 A two degree of freedom piezoelectric vibration energy harvester
with one piezoelectric element was mounted on the shaker. 195
Figure 8.10 The predicted and experimentally measured voltage output values
versus the excitation frequency for a two degree of freedom piezoelectric
vibration energy harvester inserted with one piezoelectric element. 196
Figure 8.11 The predicted and experimentally measured voltage output values
versus the external electric load resistance for a two degree of
freedom piezoelectric vibration energy harvester inserted with
one piezoelectric element. 197
Figure 8.12 The experimental setup of the two degree of freedom (2DOF)
piezoelectric vibration energy harvester (PVEH) built with two
piezoelectric elements. 198
Figure 8.13 The isolated tests for the primary and auxiliary oscillators of the two
degree of freedom (2DOF) piezoelectric vibration energy harvester
(PVEH) built with two piezoelectric elements. 199
List of Figures xviiFigure 8.14 The analytically predicted and experimentally measured voltage
outputs versus the excitation frequency for a two degree of freedom
piezoelectric vibration energy harvester inserted with two piezoelectric
elements. (A) The analytically predicted and experimentally measured
voltage output of the first piezo patch element; (B) The analytically
predicted and experimentally measured voltage output of the second
piezo patch element. 201
Figure 9.1 Statistical energy analysis model of a linear vibration energy
harvesting system. 218
Figure 9.2 Dimensionless mean harvested resonant power of linear single
degree of freedom piezoelectric and electromagnetic vibration
energy harvesters for the cases of the weak and non-weak coupling
(hM ¼ hE). 223
Figure 9.3 Resonant energy harvesting efficiency of linear single degree of
freedom piezoelectric and electromagnetic vibration energy
harvesters for the cases of weak and non-weak coupling
(hM ¼ hE). 224
Figure 10.1 Time record, autocorrelation, and autospectrum functions of
broadband random or white noise excitation input of the inertia force. 236
Figure 10.2 Time record, autocorrelation, and autospectrum functions of finite
narrow bandwidth random excitation input of the inertia force. 238
Figure 10.3 Time record, autocorrelation, and autospectrum functions of
harmonic excitation input of the inertia force. 240
Figure 10.4 Schematic for the direct methods for measurement of the mean
input and harvested power and energy harvesting efficiency of a
vibration energy harvester. DC, direct current; FFT, fast Fourier
transform; FRFP, frequency response function of power fluctuation. 242
Figure 11.1 Hydraulic actuation type of Pelamis. 250
Figure 11.2 Wave energy converter types: (A) the PS Frog point absorber;
(B) heaving buoy point absorber; (C) archimedes wave swing. 251
Figure 11.3 Rotational turbine type. 251
Figure 11.4 A single degree of freedom nonlinear oscillator in a cylindrical
tube generator. 256
Figure 11.5 Simulation schematic for Eq. (11.7) for prediction of the oscillator
relative displacement response (x-y) and output voltage v from
a sine wave base excitation acceleration input y€ at a frequency
using Matlab Simulink time domain integration method. 257
Figure 12.1 Schematic of a two degree of freedom electromagnetic vibration
energy harvester oscillator system. 272
Figure 12.2 Simulation schematic for Eq. (12.3) for prediction of the oscillator
relative displacement responses and output voltages from the base
excitation acceleration using Matlab Simulink time domain integration
method. 274
Figure 12.3 Schematic of a four degree of freedom electromagnetic vibration
energy harvester oscillator system. 275
xviii List of FiguresFigure 12.4 Schematic of an N degree of freedom electromagnetic vibration
energy harvester oscillator system. 280
Figure 12.5 Regenerative shock absorber built with a linear generator with
longitudinal or transverse magnets and highly magnetic conductive
casing. 282
Figure 12.6 Regenerative shock absorber built with the rack and pinion
mechanism. 282
Figure 12.7 Regenerative shock absorber built with the ballescrew mechanism. 283
Figure 12.8 Structure frame built with the multiple degrees of freedom
electromagnetic vibration energy harvester cells. 284
List of Figures xixList of Tables
Table 1.1 The Parameters of the Single Degree of Freedom
Spring-Mass-Dashpot System 9
Table 2.1 The Identified Single Degree of Freedom Piezoelectric System
Parameters 20
Table 3.1 Formulas for Dimensionless Harvested Resonant Power and
the Energy Harvesting Efficiency of a Piezoelectric Harvester for
Four Different Interface Circuits 55
Table 3.2 Formulas for the Peak Dimensionless Harvested Resonant Power
and Resonant Energy Harvesting Efficiency of a Piezoelectric
Harvester With Four Different Interface Circuits With the Resistance
Variation 58
Table 3.3 Formulas for the Peak Dimensionless Harvested Resonant Power
and Resonant Energy Harvesting Efficiency of a Piezoelectric
Harvester Connected With Four Different Interface Circuits With
the Force Factor Variation 59
Table 3.4 Parameters of the Single Degree of Freedom Piezoelectric Harvester
for the Case Study 60
Table 3.5 Harvested Resonant Power and Energy Harvesting Efficiency for
a Single Degree of freedom Piezoelectric Harvester Connected With
Different Interface Circuits Under a Base Acceleration of 9.8 m/s2 64
Table 4.1 Comparison of Dimensionless Harvested Power and Efficiency
for Electromagnetic Vibration Energy Harvesters Connected With
Four Different Interface Circuits 100
Table 5.1 Comparison of Electromagnetic and Piezoelectric Vibration Energy
Harvesters Connected to a Single Load Resistor 112
Table 5.2 Comparison of Electromagnetic and Piezoelectric Vibration Energy
Harvesters Connected to the Standard and SECE Interface Circuits 116
Table 5.3 Comparison of Electromagnetic and Piezoelectric Vibration Energy
Harvesters Connected to Series and Parallel SSHI Interface Circuits 117
Table 6.1 Parameters of a Quarter Vehicle Suspension Model With
Piezoelectric Insert 134
Table 7.1 System Parameters of a 2DOF Piezoelectric Vibration Energy
Harvester 161
Table 7.2 Comparison of Harvesting Performance From 1DOF to 5DOF
Piezoelectric Vibration Energy Harvester 178
Table 8.1 Piezoelectric Vibration Energy Harvester Parameters 189
Table 8.2 The Parameters of a Two Degree of Freedom Piezoelectric
Vibration Energy Harvester With One Piezoelectric Element 196
Table 8.3 The Experimentally Identified Parameters of the 2DOF Piezoelectric
Vibration Energy Harvester Built With Two Piezoelectric Elements 200
Table 9.1 Dimensionless Mean Harvested Resonant Power of Piezoelectric
and Electromagnetic Vibration Energy Harvesters at the Circuit
Oscillation Resonance 210
xxiTable 9.2 Dimensionless Mean Harvested Resonant Power of Piezoelectric
and Electromagnetic Vibration Energy Harvesters Without the
Circuit Oscillation Resonance 211
Table 9.3 Energy Harvesting Efficiency of Piezoelectric and Electromagnetic
Vibration Energy Harvesters at the Circuit Oscillation Resonance 211
Table 9.4 Energy Harvesting Efficiency of Piezoelectric and Electromagnetic
Vibration Energy Harvesters Without the Circuit Oscillation
Resonance 21
Analysis of a Single Degree
of Freedom Spring-MassDashpot System Using
Transfer Function,
Integration, State Space,
and Frequency Response
Methods
1
CHAPTER OUTLINE
1.1 Introduction . 1
1.2 Laplace Transform and Transfer Function Analysis Method 2
1.3 Time Domain Integration Method . 5
1.4 State Space Method . 6
1.5 Frequency Response Method . 8
1.6 An Example of the Time Domain Integration Simulation and Frequency Response
Analysis Methods Using Matlab Simulink Program Codes . 8
1.6.1 Time Domain Integration Simulation Method Using the Matlab Simulink
Program codes . 9
1.6.2 The Frequency Response Analysis Method Using a Matlab Code . 10
Nomenclature . 12
References . 13
Analysis of a Single
Degree of Freedom
Piezoelectric Vibration
Energy Harvester System
Using the Transfer
Function, Integration,
State Space, and
Frequency Response
Methods
2
CHAPTER OUTLINE
2.1 Introduction . 15
2.2 Analysis and Simulation of an Single Degree of Freedom Piezoelectric
Vibration Energy Harvester Connected to a Single Load Resistance . 18
2.3 Laplace Transform and Transfer Function Analysis Method 19
2.4 Time Domain Integration Method . 21
2.5 State Space Method . 23
2.6 Frequency Response Analysis Method . 26
2.7 Dimensionless Frequency Response Analysis and Harvesting Performance
Optimization . 35
Nomenclature . 40
References . 42
Analysis of Piezoelectric
Vibration Energy
Harvester System With
Different Interface Circuits
3
CHAPTER OUTLINE
3.1 Introduction . 43
3.2 Standard Interface Circuit . 44
3.3 Synchronous Electric Charge Extraction Circuit 47
3.4 Parallel Synchronous Switch Harvesting on Inductor Circuit . 49
3.5 Series Synchronous Switch Harvesting on Inductor Circuit . 52
3.6 Analysis and Comparison 54
3.7 Case Study 60
3.8 Summary . 64
Nomenclature . 65
References .
Analysis of Electromagnetic
Vibration Energy Harvesters
With Different Interface
Circuits
4
CHAPTER OUTLINE
4.1 Introduction . 69
4.2 Dimensionless Analysis of Single Degree of Freedom Electromagnetic Vibration
Energy Harvester Connected With a Single Load Resistance . 70
4.3 Laplace Transform and Transfer Function Method . 71
4.4 Time Domain Integration Method . 72
4.5 State Space Method . 74
4.6 Frequency Response Method . 75
4.7 Dimensionless Frequency Response Analysis and Harvesting Performance
Optimization . 78
4.8 Dimensionless Analysis of Electromagnetic Vibration Energy Harvesters Connected
With Energy Extraction and Storage Circuits 84
4.8.1 Standard Interface Circuit . 85
4.8.2 Synchronous Electric Charge Extraction Circuit . 88
4.8.3 Parallel Synchronous Switch Harvesting on Inductor Circuit . 91
4.8.4 Series Synchronous Switch Harvesting on Inductor Circuit . 95
4.9 Analysis and Comparison 99
4.10 Summary . 102
Nomenclature . 103
References .
Similarity and Duality
of Electromagnetic and
Piezoelectric Vibration
Energy Harvesters
5
CHAPTER OUTLINE
5.1 Introduction . 107
5.2 Dimensionless Comparison of SDOF Piezoelectric and Electromagnetic Vibration
Energy Harvesters Connected With a Single Load Resistance 108
5.3 Dimensionless Comparison of SDOF Piezoelectric and Electromagnetic Vibration
Energy Harvesters Connected With the Four Types of Interface Circuits . 115
5.4 Summary . 118
Nomenclature . 119
References
A Study of a 2DOF
Piezoelectric Vibration
Energy Harvester and
Its Application
6
H. Xiao, X. Wang
RMIT University, Melbourne, VIC, Australia
CHAPTER OUTLINE
6.1 Introduction . 123
6.2 Analysis and Simulation of a Two Degree of Freedom
Piezoelectric Vibration Energy Harvester . 125
6.3 Dimensionless Analysis of a Weakly Coupled 2DOF Piezoelectric
Vibration Energy Harvester Model 128
6.4 Case Study of a Quarter Vehicle Suspension Model and Simulation . 132
6.5 Summary . 148
Nomenclature . 150
References .
A Study of Multiple Degree
of Freedom Piezoelectric
Vibration Energy Harvester 7
H. Xiao, X. Wang
RMIT University, Melbourne, VIC, Australia
CHAPTER OUTLINE
7.1 Introduction . 155
7.2 A Two Degree of Freedom Piezoelectric Vibration Energy Harvester Inserted
With Two Piezoelectric Patch Elements . 156
7.3 A Three Degree of Freedom Piezoelectric Vibration Energy Harvester Inserted
With Three Piezoelectric Patch Elements 167
7.4 A Generalized Multiple Degree of Freedom Piezoelectric Vibration Harvester . 173
7.5 Modal Analysis and Simulation of Multiple Degree of Freedom Piezoelectric
Vibration Energy Harvester . 178
7.6 Summary . 180
Nomenclature . 181
References .
Experimental Validation of
Analytical Methods 8
H. Xiao, X. Wang
RMIT University, Melbourne, VIC, Australia
CHAPTER OUTLINE
8.1 Introduction . 187
8.2 Experimental Results of a Single Degree of Freedom Vibration Energy Harvester 188
8.2.1 Frequency Response Function . 190
8.3 Experimental Results of a Two Degree of Freedom Vibration Energy
Harvesters With One and Two Piezoelectric Elements . 194
Nomenclature .
Coupling Analysis of Linear
Vibration Energy
Harvesting Systems 9
CHAPTER OUTLINE
9.1 Introduction . 203
9.2 Coupling Analysis of a Linear Single Degree of Freedom Piezoelectric Vibration
Energy Harvesting System Under a Harmonic Excitation . 205
9.3 Coupling Analysis of a Linear Single Degree of Freedom Electromagnetic Vibration
Energy Harvesting System Under a Harmonic Excitation . 212
9.4 Coupling Analyses of Linear Piezoelectric and Electromagnetic Vibration Energy
Harvesters Under Random Excitations . 217
9.5 Relationship Between the Vibration Energy Harvesting Performance and Critical
Coupling Strength . 221
Nomenclature . 225
References
Correlation and
Frequency Response
Analyses of Input and
Harvested Power Under
White Noise, Finite
Bandwidth Random and
Harmonic Excitations
10
CHAPTER OUTLINE
10.1 Introduction . 231
10.2 Correlation and Frequency Response Analysis of Power Variables . 233
10.3 Harvested Resonant Power and Energy Harvesting Efficiency Under White Noise
Random Excitation 235
10.4 Harvested Resonant Power and Energy Harvesting Efficiency Under Finite
Bandwidth Random Excitation . 237
10.5 Harvested Resonant Power and Energy Harvesting Efficiency Under a Harmonic
Excitation 239
Nomenclature . 243
References .
Ocean Wave Energy
Conversion Analysis 11
CHAPTER OUTLINE
11.1 Introduction . 249
11.2 Analysis of a Single Degree of Freedom Nonlinear Oscillator in a Cylindrical
Tube Generator Using the Time Domain Integration Method . 254
11.3 Analysis of a Single Degree of Freedom Nonlinear Oscillator in a Cylindrical
Tube Generator Using the Harmonic Balance Method 258
11.4 Analysis of a Single Degree of Freedom Nonlinear Oscillator in a Cylindrical
Tube Generator Using the Perturbation Method . 262
Nomenclature . 265
References
Analysis of Multiple
Degrees of Freedom
Electromagnetic Vibration
Energy Harvesters and
Their Applications
12
CHAPTER OUTLINE
12.1 Analysis of a Two Degrees of Freedom Electromagnetic Vibration Energy Harvester
Oscillator System . 271
12.2 Analysis of a Four Degree of Freedom Electromagnetic Vibration Energy Harvester
Oscillator System . 276
12.3 Analysis of an N Degree of Freedom Electromagnetic Vibration Energy Harvester
Oscillator System . 279
12.4 Fields of Application . 281
Nomenclature . 285
References .
Index
‘Note: Page numbers followed by “f ” indicate figures and “t” indicate tables.’
A
Analytical methods
ANSYS modal analysis module, 188e190
bimorph cantilevered beam set up,
188, 189f
single degree of freedom piezoelectric vibration
energy harvester (SDOF PVEH), 194e195,
195f
excitation amplitudes, 187
experimental results, 200
frequency response function
alternative current (AC) voltage,
190e191
nonresonant frequencies, 191e193
predicted and measured harvested resonant
power vs. excitation frequency, 191, 192f
predicted and measured resonant output
voltages vs. excitation acceleration
amplitude, 193e194, 194f
predicted and measured resonant output
voltage vs. external electric load resistance,
193, 193f
predicted and measured voltage output vs.
excitation frequency, 191, 192f
SDOF PVEH, 194
shaker amplifier gain, 190e191
harvested resonant voltage and power, 190
Laser Vibrometer system, 195
parameters, 188, 189t, 195, 196t, 198e199, 200t
Polytec Laser Doppler Vibrometer System
display, 190, 190f
primary and auxiliary oscillators, isolated tests,
197e198, 199f
PZT-5H piezoelectric material, 187
short circuit stiffness, 188e190
two degree of freedom piezoelectric vibration
energy harvester (2DOF PVEH), 197e198,
198f
voltage outputs vs. excitation frequency,
199e200, 201f
voltage output values
vs. excitation frequency, 195e197, 196f
vs. external electric load resistance, 195e197,
197f
washers, 188, 188f
white noise random signal, 190, 191f
B
Buoy-type wave energy harvester, 252e254
C
Correlation and frequency response analyses
dynamic differential equations, 231e232
Fourier and Laplace transforms, 231e232
harvested resonant power and energy harvesting
efficiency
finite bandwidth random excitation, 237e239,
238f
harmonic excitation, 239e243, 240f, 242f
white noise random excitation, 235e237, 236f
linear vibration energy harvesting systems,
231e232
power fluctuation frequency, 232e233
power variables
autospectral densities, 233e235
cross-spectral density function, 233e235
Gaussian random variables, 233
output electric power, 233
spring-mass-dashpot system, 232e233
Coupling loss factor, 222
D
Damping loss factor, 204e205
Dimensionless harvested resonant power, 54e56,
55t
vs. dimensionless force factor, 112, 113f
vs. dimensionless resistance, 112, 114f
vs. dimensionless resistance and dimensionless
force factor, 56e57, 57f
Dimensionless mean harvested resonant power
at circuit oscillation resonance, 210e212, 210t
without the circuit oscillation resonance,
210e212, 211t
Duffing oscillator, 253e254
E
Electromagnetic and piezoelectric vibration
energy harvesters
damping coefficient, 118e119
environmental sustainability, 107
interface circuits
duality, 115e118
289Electromagnetic and piezoelectric vibration energy
harvesters (Continued)
series and parallel SSHI, 115, 117t
standard and SECE, 115, 116t
types, 115
mechanical damping, 119
single load resistance, 112t
dimensionless harvested resonant power vs.
dimensionless force factor, 112, 113f
dimensionless harvested resonant power vs.
dimensionless resistance, 112, 114f
dimensionless harvested resonant power vs.
dimensionless resistance and force factors,
110, 111f
dimensionless resonant energy harvesting
efficiency vs. dimensionless resistance and
force factors, 109e110, 109f
efficiency, 110
electric circuit oscillation resonance,
108e109
mechanical damping, 111e112
nonzero constant, 110e111
piezoelectric/magnetic and coil materials,
112e115
radial resonant frequency, 108e109
resonant energy harvesting efficiency, 108
Electromagnetic vibration energy harvester
(EMVEH)
damping coupling, 80
dimensionless analysis, 70, 71f
dimensionless frequency response analysis,
78e79
dimensionless harvested resonant power vs.
dimensionless resistance and equivalent
force factors, 81e82, 82f
efficiency vs. dimensionless resistance and force
factor, 101
electric energy extraction, 69e70
electric power generation efficiency, 80
electromechanical coupling, 99
energy extraction and storage circuits, types, 84
equivalent force factor, 71
Fourier transform, 71
frequency response method
amplitude vs. frequency, 76
base acceleration and relative oscillator
displacement, 77
definition, 75e76
displacement and voltage frequency response
functions, 76
Matlab codes, 76, 77f
open and short circuit stiffness, 77e78
root mean squared (RMS) ratio, 78
harmonic excitation, 212e217
inductor circuit
parallel synchronous switch harvesting. See
Parallel synchronous switch harvesting
series synchronous switch harvesting, 95e99,
96f
Laplace transform, 71
Matlab Simulink, 72, 72f
mechanical damping, 83
nonzero constant, 101e102
optimized power generation, 70
reference power amplitude, 83
resonant energy harvesting efficiency vs.
dimensionless resistance and equivalent
force factors, 80e81, 81f
SDOF electromagnetic harvester,
101e102, 101f
SDOF electromagneticemechanical system, 79
sinusoidal displacement excitation, 71
standard interface circuit
dimensionless harvested power, 87
efficiency, 87e88
external load resistance, 86
mechanical vibration period, 85
resonant frequency, 86
system energy equilibrium equation, 85
state space method, 74e75
storage interface circuits, 69e70
synchronous electric charge extraction (SECE)
circuit, 88e91
system optimization analyses, 99e101,
100t
time domain integration method, 72e73, 73f
transfer function method, 71
variable range limits, 84
Energy harvesting efficiency
at circuit oscillation resonance, 210e212,
211t
without the circuit oscillation resonance,
210e212, 211t
F
Fourier transform
correlation and frequency response analyses,
231e232
electromagnetic vibration energy harvester
(EMVEH), 71
piezoelectric vibration energy harvester (PVEH)
system, 26
spring-mass-dashpot system, 1e2, 4
Frequency response analysis method
290 Indexelectromagnetic vibration energy harvester
(EMVEH). See Electromagnetic vibration
energy harvester (EMVEH)
piezoelectric vibration energy harvester (PVEH)
system. See Piezoelectric vibration energy
harvester (PVEH) system
spring-mass-dashpot system, 8
H
Harvested resonant power, 25e26
piezoelectric vibration energy harvester (PVEH)
system
vs. base excitation acceleration amplitude,
25e26, 27f
vs. electrical load resistance, 25e26, 29f
vs. force factor, 25e26, 29f, 62, 63f
vs. mechanical damping, 25e26, 28f, 60e61,
62f
I
Inductor circuit
electromagnetic vibration energy harvester
(EMVEH)
parallel synchronous switch harvesting. See
Parallel synchronous switch harvesting
series synchronous switch harvesting, 95e99,
96f
piezoelectric vibration energy harvester (PVEH)
system
parallel synchronous switch harvesting. See
Parallel synchronous switch harvesting
series synchronous switch harvesting. See
Series synchronous switch harvesting
Inverse Laplace transform, 4
L
Laplace transform
correlation and frequency response analyses,
231e232
2DOF piezoelectric vibration energy harvester,
126
electromagnetic vibration energy harvester
(EMVEH), 71
piezoelectric vibration energy harvester (PVEH)
system, 19
spring-mass-dashpot system, 1e2
Linear vibration energy harvesting systems,
231e232
vs. coupling analysis
coupling loss factor, 222
critical coupling strength, 221e222
damping loss factor, 204e205
electromagnetic vibration energy harvesting
system, 212e217
electromechanical coupling factor, 204e205
environment vibrations/excitations, 203e204
force factor, 221e222
linear SDOF piezoelectric VEH, 222
moderate coupling, 223e224
normalized dimensionless force factor, 222
oscillation circuit resonance, 223
piezoelectric vibration energy harvesting
system. See Piezoelectric vibration energy
harvesting (PVEH) system
random excitations, 217e221, 218f
resonant energy harvesting efficiency, 224,
224f
statistical energy analysis (SEA), 203e204
weak and non-weak coupling, 223e224,
223f
M
Multiple degree of freedom piezoelectric vibration
energy harvester
ambient excitation frequency, 155e156
dynamic magnifier, 156
generalized model, 173e177, 177f, 178t, 179f
modal analysis, 178e180
multifrequency arrays, 156
piezoelectric patch elements
advantage, 161
configuration, 156e157, 157f
dimensionless harvested power, 166,
166fe167f
dimensionless resonant frequencies vs. mass
ratio and dimensionless frequency ratio,
156e157, 162f
dimensionless voltages, 158e160
energy harvesting efficiency, 166e167,
168fe169f
frequency difference, 163e164, 163f
governing equations, 157e158
harvesting efficiencies, 160
Laplace transform, 157e158
mass ratio, 164e165, 164f
output voltage signals, 160e161
stiffness ratio, 165e166, 165f
system parameters, 160, 161t
power density, 180e181
three degree of freedom piezoelectric vibration
energy harvester
dimensionless analysis, 171
dimensionless damping coefficient, 172,
174fe175f
Index 291Multiple degree of freedom piezoelectric vibration
energy harvester (Continued)
governing equations, 168
harvested efficiency, 172e173, 175f
Laplace transform, 169e170
mass ratio, 171, 172f
stiffness ratio, 171e172, 173f
total oscillator mass, 167, 170f
Multiple degrees of freedom electromagnetic
vibration energy harvesters
civil structures, 283e284
classification, 281
N degree of freedom electromagnetic vibration
energy harvester oscillator system,
279e280, 280f
four degree of freedom electromagnetic vibration
energy harvester oscillator system, 275f,
276e278
iron cores, 281
large vibration amplitudes, 283
regenerative shock absorbers, 284f
ballescrew mechanism, 281, 283f
linear generator, 281, 282f
rack and pinion mechanism, 281, 282f
rotational generator, 282
two degrees of freedom electromagnetic
vibration energy harvester (2DOF EMVEH)
oscillator system. See Two degrees of
freedom electromagnetic vibration energy
harvester (2DOF EMVEH) oscillator system
N
Nonlinearization methods, 253
Normalized dimensionless harvested resonant
power, 50e51
O
Ocean wave energy conversion analysis
buoy-type wave energy harvester, 252e254
dual mass system, 253
Duffing oscillator, 253e254
Halbach array, 252
nonlinearization methods, 253
Pelamis, hydraulic actuation type, 250, 250f
renewable energy sources, 249
requirements, 251e252
rotational turbine type, 250, 251f
single degree of freedom nonlinear oscillator. See
Single degree of freedom nonlinear
oscillator, cylindrical tube generator
wave energy converters (WEC), 250
types, 250, 251f
P
Parallel synchronous switch harvesting
dimensionless force factor, 52, 95
dimensionless input resonant power, 92
electric charge, 49
electromagnetic coil outgoing current, 91
harmonic base excitation, 50
load voltage, 92
mean harvested power, 50
mean harvested resonant power, 92
mean input resonant power, 50
nonzero finite number, 94
normalized dimensionless harvested resonant
power, 50e51
parallel SSHI circuit, 93
efficiency, 51
piezoelectric outgoing current, 49
Piezoelectric vibration energy harvester (PVEH)
system
base acceleration, 64, 64t
excitation, 19
dimensionless frequency response analysis
applied force/excitation acceleration, 37
dimensionless harvested resonant power vs.
dimensionless resistance and force factors,
36, 36f
electrical load resistance, 39
mechanical damping, 39e40
RC oscillation circuit, 37e38
resonant energy harvesting efficiency vs.
dimensionless resistance and force factors,
37e38, 38f
dimensionless harvested resonant power, 54e56,
55t
vs. dimensionless resistance and dimensionless
force factor, 56e57, 57f
electric energy extraction, 43, 44f, 64e65
energy harvesting efficiency, 54e56, 55t
force factor variation, 59e60, 59t
frequency bandwidth, 16e17
frequency response analysis method
base excitation acceleration amplitude,
32e33
displacement frequency response function, 31,
31f
Fourier transform, 26
harvested resonant power and output voltage
vs. frequency, 34, 34f
input excitation acceleration, 30e31, 30f
optimized electrical load resistance, 33
RungeeKuta method, 33
voltage frequency response function, 31, 32f
292 Indexharmonic excitation
blocking capacitance, 205
circuit oscillation resonance, 207e208
dimensionless mean harvested resonant power,
210e212, 210te211t
eigen equation, 206
electrical load resistance and force factor, 206
electromechanical coupling, 208e209
energy harvesting efficiency, 210e212, 211t
force factor, 205
frequency ratios, 208e209
governing equation, 205
harvested voltage ratio, 207
harvested voltage frequency response function,
206
linear SDOF system, 208, 210
mechanical damping loss factor, 209
mechanical subsystem, 205
open circuit and close circuit damping
coefficient, 208
resonance frequency, 208
resonant energy harvesting efficiency, 206
transfer function, 209
zero coupling, 207
harvested electric energy, 17e18
harvested resonant power
vs. force factor, 62, 63f
vs. mechanical damping, 60e61, 62f
inductor circuit
parallel synchronous switch harvesting. See
Parallel synchronous switch harvesting
series synchronous switch harvesting. See
Series synchronous switch harvesting
Laplace transform, 19
Matlab Simulink, 20e21, 20f
mechanical damping, 57
mechanicaleelectrical system, 17
optimization design methods, 16e17
output voltage and harvested resonant power
vs. electric resistance, 61e62, 63f
vs. input excitation acceleration, 60, 61f
parameters, 20e21, 20t, 60, 60t
relative oscillator displacement, 19
resistance variation, 58e59, 58t
resonant energy harvesting efficiency vs.
dimensionless resistance and force factors,
56, 56f
single electric load resistor, 15e16, 16f
single load resistance, 18
spring-mass oscillating system, 15e16
standard interface circuit
base excitation acceleration, 45e46
bridge rectification circuit, 44e45, 45f
dimensionless force factor, 47
mechanical vibration period, 44e45
peak efficiency, 46e47
resonant power, 46
system energy equilibrium equation,
45e46
state space method
harvested resonant power, 25e26. See also
Harvested resonant power
output sinusoidal voltage signal, 24e25, 25f
output voltage amplitude vs. electrical load
resistance, 25e26, 28f
output voltage amplitude vs. mechanical
damping, 25e26, 27f
output voltage frequency response, 24e25
output voltage vs. base excitation acceleration
amplitude, 25e26, 26f
Simulink codes, 24e25, 24f
state space equations, 23
storage interface circuits, 43, 44f, 64e65
synchronous electric charge extraction (SECE)
circuit. See Synchronous electric charge
extraction (SECE) circuit
time domain integration method. See Time
domain integration method
transfer function method, 19
weak electromechanical coupling, 54e56
Polytec Laser Doppler Vibrometer System
display, 190, 190f
Q
Quarter vehicle suspension model
analytical frequency response analysis methods,
138e139
bouncing resonant voltage magnitude, 144e146
coupling system, 146
dimensionless harvested resonant power
vs. damping ratio, 147, 149f
vs. stiffness ratio, 146, 147f
displacement amplitude ratios, 134e136, 137f
Laplace/Fourier transform, 142
Matlab program code, 143
Matlab Simulink, 134
output power, 134, 136f
output voltage, 134, 135f
vs. frequency for various quarter vehicle
masses, 144, 144f
vs. frequency for various suspension damping
coefficients, 147, 148f
vs. frequency for various suspension stiffness,
145e146, 145f
Index 293Quarter vehicle suspension model (Continued)
vs. frequency for various wheeletire damping
coefficients, 147, 148f
vs. frequency for various wheeletire mass,
143e144, 143f
vs. frequency for various wheeletire stiffness,
145e146, 145f
output voltage and harvested resonant power
vs. electric load resistance, 139e140, 139f
vs. force factor, 141e142, 142f
vs. frequency, 138e139, 138f
vs. input excitation acceleration amplitude,
137f, 138
vs. suspension damping, 140e141, 141f
vs. wheeletire damping, 140, 140f
parameters, 132, 134t
and simulation, 132, 133f
sine wave base excitation input and sinusoidal
voltage output, 134, 135f
time domain simulation, 138e139
R
Regenerative shock absorbers, 284f
ballescrew mechanism, 281, 283f
linear generator, 281, 282f
rack and pinion mechanism, 281, 282f
RungeeKuta method, 33
S
Series synchronous switch harvesting, 95e99,
96f
dimensionless force factor, 54
electrical quality factor, 52
equivalent load resistance, 52
mean harvested power, 53
mechanical vibration period, 52
rectified voltage, 52
SSHI circuit efficiency, 53e54
Single degree of freedom (SDOF)
piezoelectric vibration energy harvester (PVEH)
system. See Piezoelectric vibration energy
harvester (PVEH) system
spring-mass-dashpot system
displacement frequency response function, 4
Fourier transform, 1e2, 4
frequency response method, 8
inertial force, 3, 3f
inverse Laplace transform, 4
Laplace transform, 1e2
Matlab Simulink transfer function method,
4e5, 5f
normalized conversion, 2
parameters, 8, 9t
“shunt” damping, 1e2
state space method, 6e7, 8f
time domain integration method. See Time
domain integration method
transducer mechanism, 2
weak coupling, 2
Single degree of freedom nonlinear oscillator,
cylindrical tube generator
harmonic balance method, 258e261
perturbation method, 262e265
time domain integration method, 256f
excitation frequencies and amplitude, 257
Gaussian meter, 257
Matlab Simulink, 256e257, 257f
Newton’s second law, 255e256
nonlinearity, types, 254e255
state space analysis methods, 258
transfer function, 258
State space method
electromagnetic vibration energy harvester
(EMVEH), 74e75
piezoelectric vibration energy harvester (PVEH)
system. See Piezoelectric vibration energy
harvester (PVEH) system
spring-mass-dashpot system, 6e7, 8f
Statistical energy analysis (SEA) model,
203e204, 218, 218f
Synchronous electric charge extraction (SECE)
circuit, 88e91
dimensionless force factor, 49
dimensionless harvested resonant power, 48
efficiency, 48
mean harvested power, 47
mechanical velocity, 47
T
Time domain integration method, 72e73, 73f
excitation acceleration, 21e22
Matlab Simulink program codes, 11f
frequency response analysis, 10e12, 11fe12f
sine wave excitation acceleration, 9e10, 10f
sine wave excitation signal, 9e10, 9f
relative acceleration, 21e22, 22f
relative displacement frequency response, 5, 6f
single degree of freedom nonlinear oscillator,
256f
excitation frequencies and amplitude, 257
Gaussian meter, 257
Matlab Simulink, 256e257, 257f
Newton’s second law, 255e256
nonlinearity, types, 254e255
294 Indexstate space analysis methods, 258
transfer function, 258
Two degree of freedom piezoelectric vibration
energy harvester (2DOF PVEH),
197e198, 198f
advantages, 123e124
electrical system governing equation, 125e126
force factor and blocking capacitance, 126
frequency response analysis method, 149e150
harmonic excitation, 127
Laplace transform, 126
mechanical system governing equations, 125,
125f
microelectromechanical system (MEMS)
applications, 124
output voltage magnitude, 127
piezoelectric materials, 123e124
quarter vehicle suspension model, 125. See also
Quarter vehicle suspension model
resonant frequency tuning structure, 124
transfer function equations, 126
tuned auxiliary structure, 124
weakly coupled model, 128e131, 132fe133f
Two degrees of freedom electromagnetic vibration
energy harvester (2DOF EMVEH)
oscillator system
coil oscillators, 271e273, 272f
damping coefficients, 271e273
electromechanical system governing equations,
271e273
Laplace transform, 273
Matlab program codes, 273
relative displacement responses and output
voltages, 274f, 275
W
Wave energy converters (WEC), 250, 251f
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